In Figure 5 we plot the distributions of these SDSS-WISE quasars and stars in the
z-W1 versus \(g-z\)  diagram. Obviously, most quasars with redshifts less than four can be
separated from both normal and late-type stars on this color-color diagram.
This is also confirmed by the median z-W1 and \(g-z\)  colors at different redshift,
shown as yellow solid line in Figure 5, which is obtained from the median color-redshift
relation of SDSS-WISE quasars. However,
there are significant overlaps, especially between quasars with \(z>4\)  and stars.
Similar to in [1]}, we perform an automatic search for the best criterion
to efficiently separate quasars and stars. We obtain this
criterion as: \(z-W1=0.66(g-z)+2.01\) . With this criterion, we can select 36895 of
37,535 quasars (with a percentage of 98.30%) and select 33,442 of 35,124 stars (with a percentage of 95.21%). The false positive rate, defined as the ratio between the number
of stars (1682) incorrectly selected as quasars and the number of all sources selected as
quasars (38,577) by our criterion, is 4.36%. However, we must keep in mind that the actual number of
stars could be significantly larger than what we used here because only a tiny fraction of stars have been
spectroscopically observed by SDSS. Therefore, the real false positive rate of our quasar selection may be higher.
For 37,272 SDSS-WISE quasars with redshifts less than four, with the proposed criterion
we can select 36,827 of them, with completeness of 98.81%.
This demonstrates the very high efficiency
in selecting \(z<4\)  quasars with the \(z-W1/g-z\)  criterion, similar to using the
\(Y-K/g-z\)  criterion [1]}.
